91Ó°ÊÓ

Color Rugs is holding a two-week carpet sale at Jerry's Club, a local warehouse store. Color Rugs plans to sell carpets for \(\$ 500\) each. The company will purchase the carpets from a local distributor for \(\$ 350\) each, with the privilege of returning any unsold units for a full refund. Jerry's Club has offered Color Rugs two payment alternatives for the use of space. Option 1: A fixed payment of \(\$ 5,000\) for the sale period Option \(2: 10 \%\) of total revenues earned during the sale period Assume Color Rugs will incur no other costs. 1\. Calculate the breakeven point in units for (a) option 1 and (b) option 2 . 2\. At what level of revenues will Color Rugs earn the same operating income under either option? a. For what range of unit sales will Color Rugs prefer option \(1 ?\) b. For what range of unit sales will Color Rugs prefer option \(2 ?\) 3\. Calculate the degree of operating leverage at sales of 100 units for the two rental options. 4\. Briefly explain and interpret your answer to requirement 3.

Step by step solution

01

Define the Contribution Margin per Unit

The contribution margin per unit is calculated as the selling price minus the cost price. \[ \text{Contribution Margin per Unit} = \text{Selling Price} - \text{Cost Price} = \\(500 - \\)350 = \$150 \]

02

Calculate Breakeven Point for Option 1

To find the breakeven point for Option 1, where there is a fixed fee of \\(5,000, we use the formula: \[ \text{Breakeven Point (Units)} = \frac{\text{Fixed Costs}}{\text{Contribution Margin per Unit}} = \frac{\\)5,000}{\$150} \approx 33.33 \] Since we can't sell a fraction of a carpet, round up to 34 units.

03

Calculate Breakeven Point for Option 2

For Option 2, where 10% of revenues is paid, we set up the equation: \[ \text{Revenues} = \text{Units Sold} \times \text{Selling Price} = x \times \\(500 \] Costs include 10% of revenue, so:\[ 0.10 \times \text{Revenues} = 0.10 \times (x \times \\)500) = 0.10 \times x \times \\(500 \] The breakeven point satisfies: \[ x \times \\)150 = 0.10 \times x \times \\(500 \] Solving gives: \[ x = \frac{0.10 \times \\)500}{\\(150 - 0.10 \times \\)500} = 11.11 \] Round up to 12 units.

04

Find Revenue Level for Equal Operating Income

To find when operating incomes are equal, equate both operating income formulas: \[ x \times \\(150 - \\)5,000 = 0.90 \times x \times \$500 \]Solving gives:\[ x = 45.45 \]Round to 46 units.

05

Determine Preference Range for Option 1

Color Rugs prefers Option 1 when selling fewer than 46 units because fixed costs lead to lower total expenses than Option 2.

06

Determine Preference Range for Option 2

Color Rugs prefers Option 2 when selling more than 46 units because the percentage cost becomes more economical compared to the fixed cost.

07

Calculate Degree of Operating Leverage (DOL) at 100 Units for Option 1

Degree of Operating Leverage (DOL) is calculated as \[ \text{DOL} = \frac{x \times \text{CM}}{x \times \text{CM} - \text{Fixed Costs}} = \frac{100 \times \\(150}{100 \times \\)150 - \$5,000} = \frac{15,000}{10,000} = 1.5 \]

08

Calculate Degree of Operating Leverage (DOL) at 100 Units for Option 2

Degree of Operating Leverage (DOL) for Option 2 is \[ \text{DOL} = \frac{x \times \text{CM} }{x \times \text{CM} - 0.10 \times x \times \\(500} = \frac{100 \times \\)150}{100 \times (\\(150) - 100 \times 0.10 \times \\)500} = 1.111 \]

09

Interpret DOL

A higher DOL, as shown in Option 1, indicates more sensitivity to changes in sales levels due to fixed costs impacting profits significantly compared to Option 2's more variable cost structure.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Breakeven Analysis

Breakeven analysis is a fundamental concept in cost accounting, crucial for understanding when a business will start making a profit. It helps identify the point where revenues equal the total costs, meaning there is no net loss or gain. For Color Rugs, two scenarios show how breakeven analysis applies. In option 1, with a fixed payment, the breakeven point is determined by dividing the fixed costs by the contribution margin per carpet. This results in needing to sell 34 carpets to cover the $5,000 charge. In option 2, costs fluctuate with revenue since 10% of revenue is a payment. This situation changes the breakeven formula. Instead of fixed expenses, now 10% of sales is the cost, leading to a lower breakeven point of 12 carpets because each additional sale increases costs by 10%. Breakeven analysis helps businesses decide which option minimizes risk or maximizes profit based on sales predicts.

Contribution Margin

The contribution margin is an essential tool for assessing how much each sold unit contributes to covering fixed costs and generating profit. It is the difference between the selling price and the unit variable costs. For Color Rugs: - Selling price per carpet: $500 - Purchase cost per carpet: $350 The contribution margin per carpet is thus $150 This means every carpet sold provides a $150 buffer to offset fixed expenses and eventually contribute to profit once all fixed costs are covered. The contribution margin assists in figuring out the breakeven point by indicating how much sale revenue is available per unit to cover fixed costs. Understanding this concept is key for management to make informed decisions on pricing, cost control, and sales strategies.

Degree of Operating Leverage

Operating leverage refers to how sensitive net income is to a change in sales volume. It highlights the weight of fixed costs in a company's cost structure. A high degree of operating leverage (DOL) indicates that a small change in sales can lead to a large change in operating income. For Color Rugs, the DOL is calculated at a sales volume of 100 units. In option 1, the DOL is 1.5, indicating high sensitivity due to fixed space rental costs. A modest increase in sales would dramatically improve profits once fixed costs are covered. In option 2, the DOL is lower at 1.111 because costs are more variable, and profits move less sharply with sales changes. Understanding the degree of operating leverage aids businesses in determining variability in their operating income as they scale operations.

Fixed and Variable Costs

Fixed costs are constant regardless of sales volume. Here, it's a $5,000 fee for the sales space in option 1. These costs do not vary with the number of units sold and must be paid regardless of revenue. The presence of high fixed costs can mean higher risk, as they need to be covered even if sales decrease. Conversely, variable costs change with production volume. In option 2, the cost is 10% of revenue, making it a variable cost. This means the overall cost fluctuates with sales volumes, offering more flexibility and potentially lower risk at lower sales volumes since costs decrease in line with revenues. The distinction between fixed and variable costs is crucial for budgeting and financial planning. It helps businesses understand how costs will behave as production and sales levels change, informing strategic decisions on pricing, market entry, and cost control.